66 



KNOWLEDGE. 



[March 1, 1900. 



how vast a region of space we consider, the yield of light 

 cannot exceed a definite amount? We ai'e faced in fact 

 with the question which the machinery of the integral 

 calculus has been constructed to answer. What is the 

 value of infinity multiplied by zero? Such a product 

 IS called by mathematicians an indeterminate form ; 

 indeterminate since its value cannot be ascertained 

 until we know by what process the infinitely great 

 was reached, and likewise the infinitely small. In the 

 particular case before us, the infinitely great was 

 reached by receding further and further into space, 

 which also brought about the infinite smallncss of each 

 individual star. A connection clearly exists between 

 the growth in number and the diminution in size, and 

 this connection entirely depends on the manner in which 

 the stars are distributed throughout space. This is a point 

 which Mr. Maunder has made, and it cannot be too 

 strongly emphasized. Without considering first the 

 method of distribution, it is perfectly futile to attack 

 the problem, in fact there is no problem to attack. We 

 must first postulate a particular formation, and then, 

 and only then, can we apply rigid mathematical 

 reasoning. 



The formation which naturally occurs to us first, is 

 that of a uniform distribution. This distribution may 

 be obtained by considering the stars distributed on con- 

 centric spheres whose radii increase by equal steps. 

 Then the number of stars on any one spherical arrange- 

 ment would vai-y as the area of the sphere, but the 

 apjjarent disc of each star would vary inversely as the 

 area of the sphere, since it varies inversely as the square 

 of the radius. Thus the growth in number would 

 exactly balance the decrease in size, and the luminous 

 area contributed by each spherical distribution would 

 be the same. This contribution is c^uite finite, therefore 

 by taking enough distributions we can obtain any 

 desired yield of light, enough for instance to completely 

 fill *]ie heavens. The above argument is quite inde- 

 pendent of the size or distance apart of the stars, and it 

 leads us to the very interesting fact, that granted the 

 perfect transmissibility of the ether, and non-inter- 

 ference by dark bodies, then no matter how diffused 

 the stars are through space, so long as that distribution 

 is maintained to an infinite distance in all directions, thj 

 appearance of the heavens ought to be complete bright- 

 ness. All we require is that at no region of space shall 

 the density of star distribution become indefinitely 

 small. It may fluctuate, but it must not become ever 

 indefinitely small. Assuming that stars are a million 

 miles in diameter, and spaced uniformly twenty billion 

 miles apart, I find that a region of space, ten thousand 

 trillion miles in radius, would be sufficient to com- 

 pletely fill the heavens with light to an observer 

 at the centre. 



Let us now take the case where the stars thin out in 

 numbers as the distance from the eai-th increases. In 

 order to give numerical results, let us again assume that 

 the stars are all a million miles in diameter, and that 

 those nearest the earth are spaced twenty billion miles 

 apart. If the number of stai-s per unit volume of space 

 varies inversely as any positive power of the radius, we 

 get a distribution which progressively thins out. We 

 have just dealt with the case in which this index power 

 is zero. When powers other than zero are assumed, the 

 problem resolves itself into a simple case of integration. 

 As the powers increase from zero upwards, the rate at 

 which the density of distribution falls off increases, and 

 consequently a greater and greater region of space must 

 be included in order to block every direction with a 



star. When the index power is uni+y, or in other words 

 when the density of distribution varies inversely as the 

 radius, this distance is so stupendously great that I am 

 almost afraid to mention it. One followed by twenty 

 billion noughts, and then multiplied by twenty billion, 

 will almost suffice. It is clear we are nearing the law 

 of disti'ibution when it will be necessary to include an 

 infinite region of space in order to occupy all directions 

 with a star. This stage is reached when the index of 

 the power exceeds unity by only one five hundred 

 billionth. If the law of distribution gives an ever so 

 slightly quicker rate of thinning out, we cannot, even 

 by considering an infinite region of sjiace, gather up 

 enough to make the heavens a complete blaze of light. 



In fact, all infinite distributions fall into two 

 classes : — • 



(1) Those in which all directions are blocked by stars, 



(2) Those in which interstellar spaces exist ; 



And the law of distribution which fonns a link be- 

 tween these two classes, occurs when the power of the 

 radius very slightly exceeds unity. 



When the power is two, interstellar spaces so vastly 

 exceed the luminous area, that only about one forty 

 billionth of the heavens is illuminated. It is very in- 

 structive to note how an exceedingly small change in 

 the law of distribution produces an exceedingly great 

 difference in the amount of the heavens illuminated. 



There is another question which should be considered 

 in connection with this subject. When we assume a 

 progressively decreasing density of star distribution, 

 might not the same reason, which in some cases makes 

 it impossible to obtain more than a finite amount of 

 illumination, also prevent us from obtaining more than 

 a finite number of stai-s ? The answer is, that if the 

 density varies inversely as the radius raised to a power 

 greater than three, then indeed the number of stars in 

 infinite space will be finite. With a less rapid thinning 

 out the number is infinite. 



We have seen that the distances dealt with in con- 

 nection with this problem are absolutely stupendous; 

 so great, in fact, that the life of a star- would be a mer^i 

 nothing compai'ed with the length of time occujiied by 

 its light in coming to us. This brings up the considera- 

 tion that if such vast regions of space are to be 

 considered then the probability of any one direction 

 being occupied by a brigat star is exceedingly remote. 

 In fact, dark stars ought to outnumber bright stars by 

 millions or billions to one. Thi«i being true, at first 

 glance one would expect that eclipses of bright suns 

 would be a -common occurrence. But a moment's re- 

 flection shows, that as far as the visible stars are con- 

 cerned, the interstellar spaces are so vastly greater than 

 the spaces occupied by bright surs, that dark stars 

 might outnumber them by billions to one without 

 making an eclipse probable. It would only be in the 

 more distant depths of s,)ace, depths so profound that 

 our most mighty telescopes could never identify single 

 stai-s therein, that the blocking out of light by the 

 dark bodies would have effect. For instance, in the 

 homogeneous distribution we first considered, although 

 all directions would be blocked by stai-s, yet it may be, 

 only an infinitesimal proportion of these are bright 

 stars. This theoi-y for accounting for the black back- 

 ground to the bright stars has not, I think, been 

 brought up in the recent interesting discussion, and it 

 seems to me to be worthy of a place in Mr. Burns' list 

 of possible hypotheses. 



Charles E. Inglis, b.a. 



Kings College, Cambridge. 



